slides chap2

Qubit: Definition & Properties Pauli Matrices & 2D Operators Quantum Gates Composite systems Physical Realization

Qubit: The Unit of Quantum Computation

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Lecture 1

Qubit, Pauli matrices, 2D operators

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Qubit: The Unit of Quantum Computation1 Qubit: Definition & Properties

What is a qubit?Bloch Sphere Representation

2 Pauli Matrices & 2D OperatorsIntroductionPropertiesOperators in 2D Hilbert Space

3 Quantum GatesGatesExamples√NOT Gate

Decomposition of Single-qubit Gates4 Composite systems

Product StatesMultiple QubitsControlled Gates

5 Physical Realization 3 / 23


Qubit

The bit (binary digit) is the basic unit of classicalcomputation, having two values – 1 (on) and 0 (off); atwo-state device, e.g., a transistor

The quantum bit is the unit of a quantum computer. It is atwo-level quantum system, represented by a vector in 2DHilbert Space, whose basis states can be labelled as

|0〉 =

(10

), |1〉 =

(01

)analogous to the on and off states of a classical bit. Physicalexamples of qubit: an electron spin (along a given direction)with two eigenstates | ↑〉 and | ↓〉, or vertical and horizontalpolarization states of a photon, or the ground and excitedstates of an atom.

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Qubit...

The basis {|0〉, |1〉} is known as the standard basis or thecomputational basis in 2D Hilbert space.

The most general state of a qubit is |ψ〉 = α|0〉+ β|1〉. Thequbit can exist in a superposition of on and off states, unlikea classical bit. This property is of paramount importance toquantum computing.

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Bloch Sphere Representation

For a normalized qubit, |α|2 + |β|2 = 1. This suggests arepresentation in spherical polar coordinates,

|ψ〉 = e−iφ

2 cos(θ

2)|0〉+ e

iφ2 cos(

θ

2)|1〉, (0 ≤ θ ≤ π; 0 ≤ φ ≤ 2π)

This is called the Bloch Sphere Representation

qubit is thus a point onthe surface of a sphere ofunit radius, parametrizedby the coordinates (θ, φ).

The basis states |0〉 and|1〉 are then represented bythe north and south polesof the sphere, respectively.

Figure : Qubit on Bloch sphere.

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Pauli Matrices

Any quantum system in 2D Hilbert Space, e.g., an electron spin,can be adequately described by the 3 Pauli matrices:

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

)

The corresponding electron spin operators areSx = ~

2σx , Sy = ~2σy , Sz = ~

2σz .

The computational basis states, |0〉 and |1〉 are theeigenstates of σz , corresponding to spin-up and spin-downstates along z-direction.

The eigenstates of σx are, |+〉 = 1√2

(|0〉+ |1〉) and

|−〉 = 1√2

(|0〉 − |1〉).

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Pauli Matrices: Properties

1 σx , σy , σz are Hermitian and Unitary, i.e., σ†i σi = σ2i = I.

2 σx , σy , σz have two eigenvalues, ±1.

3 det(σi ) = 1.

4 Trace(σi) = 0.

5 σiσj = iεijkσk ∀ i 6= j .

6 From (5), it follows that [σi , σj ] = 2iεijkσk if i 6= j .

7 From (1) and (5), the anti-commutator{σi , σj} = σiσj + σjσi = δij .

8 Also, (1) and (5) can be condensed into σiσj = δij I + εijkσk9 From (1) and (6), for a unit vector n̂,

(n̂ · σ)2 = (nx · σx + ny · σy + nz · σz)2 = I

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2D Operators

The three σi and I2 form an independent set.

Any 2× 2 unitary matrix has four independent parameters.

The Pauli matrices, along with I2 form a complete basis in thespace of 2× 2 unitary matrices.

Therefore, any operator in 2D Hilbert Space can be written downas:

U =3∑

i=1

ciσi + c4I

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Rotation Operator

The operator U[R(θ)] = e−i~θ·σ/2 causes rotations in 2D Hilbert

Space. On expanding the exponential and grouping terms, we find:

e−i~θ·σ/2 = cos

(θ

2

)I− i sin

(θ

2

)(θ̂ · σ)

This will prove very useful later on for constructing variousquantum gates.

Example

Take the operator for rotation along x-axis, U[Rx(θ)] = e−iθσx/2

e−iθσx/2 =

(cos(θ/2) −i sin(θ/2)−i sin(θ/2) cos(θ/2)

)Using the Bloch sphere, examine its action on the |0〉, |1〉 states.

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Lecture 2

Quantum gates, decomposition of single-qubit gates

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Quantum Gates

In classical computation, logic operations on bitsare represented by gates, e.g., the NOT gateflips a bit:

Input Output

0 11 0

Quantum gates are quite clearly unitary operators.

As we have already seen, all single qubit gates can berepresented in terms of Pauli matrices and the identity matrix.

The quantum analogue of the NOT gate is σx .

σx |0〉 =

(0 11 0

)(10

)=

(01

)= |1〉; σx |1〉 = |0〉

Since for every U there is a U†, all quantum gates arereversible.

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Quantum Gates...

Quantum mechanics allows construction of gates that have noclassical analogue. Two important single qubit gates are:

the phase gate,

S =

(1 00 i

); S

(|0〉+ |1〉√

2

)=|0〉+ i |1〉√

2

which introduces a relative phase of π,

the Hadamard gate,

H =1√2

(1 11 −1

); H|0〉 =

|0〉+ |1〉√2

, H|1〉 =|0〉 − |1〉√

2

which transforms the computational basis to the |±〉 statesand vice versa

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√NOT Gate

Another example of a uniquely quantum gate is the√NOT gate.

To construct this, we note that:

1

√NOT

√NOT = σx

2 σx =

(0 11 0

)= iU[Rx(π)]

Thus, up to an overall phase, which is of no importance whilecalculating probabilities,

√NOT = U[Rx(π/2)] =

1√2

(1 −i−i 1

)

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Decomposition of Single-qubit Gates

In the derivation of√NOT gate, we expressed the NOT gate

as a rotation operator. More generally, any single qubit unitaryoperation can be written as U = e iαRn̂(θ), where n̂ is the axisand θ is the angle of rotation, and e iα is a global phase shift.

Any single qubit gate can be decomposed into variouscombinations of rotations and global phase shifts. Particularlyimportant is the Z-Y decomposition:

U = e iαRz(β)Ry (γ)Rz(δ)

Analogous decompositions can be carried out in terms ofrotations along any two non-parallel axis n̂ and m̂:U = e iαRn̂(β)Rm̂(γ)Rn̂(δ)

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Decomposition of Single-qubit Gates...

Using the Z-Y decomposition, it can be shown that for anyunitary U we can find unitary A, B, C such that ABC = Iand U = e iαAσxBσxC

Example (Decomposition of Hadamard)

H = Rz(3π)Ry (π

2)Rz(π) (Verify)

1√2

(1 11 −1

)=

(i 00 −i

)(cos(π/8) − sin(π/8)sin(π/8) cos(π/8)

)σx (1)(

cos(π/8) sin(π/8)− sin(π/8) cos(π/8)

)(−1 00 −1

)σx

(i 00 −i

)

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Lecture 3

Composite systems, product representation, controlled gates,physical realization

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Product States

For computational purposes we require more than one qubit. Tostudy multi-qubit states, the tensor-product notation is useful.

Suppose you have two systems A and B in states |ψ1〉A and|ψ2〉b respectively. The state of the composite system A+Bcan be written in the form |ψ1〉A⊗ |ψ2〉B where ⊗ denotes thetensor product.

In matrix notation, product states are expressed as follows. If

〈ψ|1 =(a1 a2 a3 · · · an

)〈φ|2 =

(b1 b2 b3 · · · bn

)Then

〈ψφ| = 〈ψ|1 ⊗ 〈φ|2=

(a1b1 a1b2 · · · a1bn a2b1 a2b2 · · · a2bn

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Multiple Qubits

States with n qubits can have states of the form |000...00〉,|111...11〉, |010...00〉 etc. and their superpositions. The states

1√n⊗n

i=1,xi∈{0,1}|xi 〉

form the computational basis in a n-qubit system.

Example (2-qubit system)

The states |00〉, |01〉, |10〉, |11〉 form the computational basis fora 2-qubit system.

|00〉 =

(10

)⊗(

10

)=

1000

, |01〉 =

(10

)⊗(

01

)=

0100

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The CNOT gate

The CNOT or controlled-NOT gate is a 2-qubit gate that flipsthe 2nd qubit only if the first qubit is in the |1〉 state.

Action of CNOT on 2-qubit computational basis

|00〉 CNOT−−−−→ |00〉, |01〉 CNOT−−−−→ |01〉

|10〉 CNOT−−−−→ |11〉, |11〉 CNOT−−−−→ |10〉

Equivalently, |A,B〉 CNOT−−−−→ |A,A⊕ B〉 where ⊕ denotes additionmodulo 2.

The matrix representation ofCNOT is

1 0 0 00 1 0 00 0 0 10 0 1 0

The circuit diagram is

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Controlled Gates

The CNOT is an example of a broad class of controlled gates thatuse a single qubit as control and act only if the control is set to |1〉.

A controlled- U gate is represented in

matrix form as

(I2 00 U

)where U

can be an operator of any dimension.

Figure : Controlled-U

Note

Any arbitrary gate can be constructed using only single-qubitoperations and CNOT.

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Physical Realization of Qubits

As mentioned before, any two-level quantum system can be aqubit.

In NMR quantum computers, nuclear spins are used as qubitswith spin-up and spin-down states as the computational basis.Gates are applied by magnetic pulses on these nuclear spins.

In superconducting phase qubit, the direction (phase) ofcurrent (clockwise or anticlockwise) is the quantum state.

In optical quantum computers, the |0〉 and |1〉 states aredefined as the states of photons being in either of twopotential wells.

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References

M.A. Nielsen & I.L. Chuang: Quantum Computation andQuantum Information, Cambridge University Press, ISBN9781107002173

John Preskill, Lecture Notes on Quantum Computation,http://www.theory.caltech.edu/~preskill/ph219/

David Deutsch, Lectures on Quantum Computation

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http://www.theory.caltech.edu/~preskill/ph219/

http://www.quiprocone.org/Protected/DD_lectures.htm

Documents

slides chap2